Quality of model and Error Analysis in Variational Data Assimilation François-Xavier LE DIMET Victor SHUTYAEV Université Joseph Fourier+INRIA Projet IDOPT, Grenoble, France Russian Academy of Sciences Institute of Numerical Mathematiques

Prediction: What information is necessary ? Model Model - law of conservation mass, energy - Laws of behaviour - Parametrization of physical processes Observations in situ and/or remote Observations in situ and/or remote Statistics Statistics Images Images

Forecast.. Produced by the integration of the model from an initial condition Produced by the integration of the model from an initial condition Problem : how to link together heterogeneous sources of information Problem : how to link together heterogeneous sources of information Heterogeneity in : Heterogeneity in : Nature Nature Quality Quality Density Density

Basic Problem U and V control variables, V being and error on the model U and V control variables, V being and error on the model J cost function J cost function U* and V* minimizes J U* and V* minimizes J

Optimality System P is the adjoint variable. P is the adjoint variable. Gradients are couputed by solving the adjoint model then an optimization method is performed. Gradients are couputed by solving the adjoint model then an optimization method is performed.

Errors On the model On the model Physical approximation (e.g. parametrization of subgrid processes) Physical approximation (e.g. parametrization of subgrid processes) Numerical discretization Numerical discretization Numerical algorithms ( stopping criterions for iterative methods Numerical algorithms ( stopping criterions for iterative methods On the observations On the observations Physical measurement Physical measurement Sampling Sampling Some « pseudo-observations », from remote sensing, are obtained by solving an inverse problem. Some « pseudo-observations », from remote sensing, are obtained by solving an inverse problem.

Sensitivity of the initial condition with respect to errors on the models and on the observations. The prediction is highly dependant on the initial condition. The prediction is highly dependant on the initial condition. Models have errors Models have errors Observations have errors. Observations have errors. What is the sensitivity of the initial condition to these errors ? What is the sensitivity of the initial condition to these errors ?

Optimality System : including errors on the model and on the observation

Second order adjoint

Models and Data Is it necessary to improve a model if data are not changed ? Is it necessary to improve a model if data are not changed ? For a given model what is the « best » set of data? For a given model what is the « best » set of data? What is the adequation between models and data? What is the adequation between models and data?

A simple numerical experiment. Burger’s equation with homegeneous B.C.’s Burger’s equation with homegeneous B.C.’s Exact solution is known Exact solution is known Observations are without error Observations are without error Numerical solution with different discretization Numerical solution with different discretization The assimilation is performed between T=0 and T=1 The assimilation is performed between T=0 and T=1 Then the flow is predicted at t=2. Then the flow is predicted at t=2.

Partial Conclusion The error in the model is introduced through the discretization The error in the model is introduced through the discretization The observations remain the same whatever be the discretization The observations remain the same whatever be the discretization It shows that the forecast can be downgraded if the model is upgraded. It shows that the forecast can be downgraded if the model is upgraded. Only the quality of the O.S. makes sense. Only the quality of the O.S. makes sense.

Remark 1 How to improve the link between data and models? How to improve the link between data and models? C is the operator mapping the space of the state variable into the space of observations C is the operator mapping the space of the state variable into the space of observations We considered the liear case. We considered the liear case.

Remark 2 : ensemble prediction To estimate the impact of uncertainies on the prediction several prediction are performed with perturbed initial conditions To estimate the impact of uncertainies on the prediction several prediction are performed with perturbed initial conditions But the initial condition is an artefact : there is no natural error on it. The error comes from the data throughthe data assimilation process But the initial condition is an artefact : there is no natural error on it. The error comes from the data throughthe data assimilation process If the error on the data are gaussian : what about the initial condition? If the error on the data are gaussian : what about the initial condition?

Because D.A. is a non linear process then the initial condition is no longer gaussian

Control of the error

Choice of the base

Remark. The model has several sources of errors The model has several sources of errors Discretization errors may depends on the second derivative : we can identify this error in a base of the first eigenvalues of the Laplacian Discretization errors may depends on the second derivative : we can identify this error in a base of the first eigenvalues of the Laplacian The systematic error may depends be estimated using the eigenvalues of the correlation matrix The systematic error may depends be estimated using the eigenvalues of the correlation matrix

Numerical experiment With Burger’s equation With Burger’s equation Laplacian and covariance matrix have considered separately then jointly Laplacian and covariance matrix have considered separately then jointly The number of vectors considered in the correctin term varies The number of vectors considered in the correctin term varies

With the eigenvectors of the Laplacian

Model error estimation controlled system model cost function optimality conditions adjoint system(to calculate the gradient)

Reduction of the size of the controlled problem Change the space bases Suppose is a base of the phase space and is time-dependent base function on [0, T], so that then the controlled variables are changed to with controlled space size

Optimality conditions for the estimation of model errors after size reduction If P is the solution of adjoint system, we search for optimal values of to minimize J :

Problem : how to choose the spatial base ? Consider the fastest error propagation direction Amplification factor Choose as leading eigenvectors of Calculus of - Lanczos Algorithm

Numerical experiments with another base Choice of “correct” model : - fine discretization: domain with 41 times 41 grid points To get the simulated observation - simulation results of ‘ correct’ model Choice of “incorrect” model : - coarse discretization: domain with 21 times 21 grid points

The difference of potential field between two models after 8 hours’ integration

Experiments without size reduction (1083*48) : the discrepancy of models at the end of integration before optimization after optimization

Experiments with size reduction (380*48) : the discrepancy of models at the end of integration before optimization after optimization

Experiments with size reduction (380*8) : the discrepancy of models at the end of integration before optimization after optimization

Conclusion For Data assimilation, Controlling the model error is a significant improvement. For Data assimilation, Controlling the model error is a significant improvement. In term of software development it’s cheap. In term of software development it’s cheap. In term of computational cost it could be expensive. In term of computational cost it could be expensive. It is a powerful tool for the analysis and identification of errors It is a powerful tool for the analysis and identification of errors